Solution to the second-order Sylvester matrix equation MVF2+DVF+KV=BW

This note provides a complete general parametric solution (V,W) to the generalized second-order Sylvester matrix equation MVF²+DVF+KV=BW, with F being an arbitrary square matrix. The primary feature of this solution is that the matrix F does not need to be in any canonical form, or may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems involving second-order dynamical systems.